1 | # pylint: disable=line-too-long |
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2 | r""" |
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3 | Definition for 2D (orientated system) |
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4 | ------------------------------------- |
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5 | |
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6 | The angles $\theta$ and $\phi$ define the orientation of the axis of the |
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7 | cylinder. The angle $\Psi$ is defined as the orientation of the major |
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8 | axis of the ellipse with respect to the vector $Q$. A gaussian polydispersity |
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9 | can be added to any of the orientation angles, and also for the minor |
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10 | radius and the ratio of the ellipse radii. |
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11 | |
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12 | .. figure:: img/elliptical_cylinder_geometry.png |
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13 | |
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14 | Elliptical cylinder geometry $a = r_\text{minor}$ |
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15 | and $\nu = r_\text{major} / r_\text{minor}$ is the *axis_ratio*. |
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16 | |
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17 | The function calculated is |
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18 | |
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19 | .. math:: |
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20 | |
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21 | I(\vec q)=\frac{1}{V_\text{cyl}}\int{d\psi}\int{d\phi}\int{ |
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22 | p(\theta,\phi,\psi)F^2(\vec q,\alpha,\psi)\sin(\alpha)d\alpha} |
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23 | |
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24 | with the functions |
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25 | |
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26 | .. math:: |
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27 | |
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28 | F(q,\alpha,\psi) = 2\frac{J_1(a)\sin(b)}{ab} |
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29 | |
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30 | where |
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31 | |
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32 | .. math:: |
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33 | |
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34 | a = qr'\sin(\alpha) |
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35 | |
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36 | b = q\frac{L}{2}\cos(\alpha) |
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37 | |
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38 | r'=\frac{r_{minor}}{\sqrt{2}}\sqrt{(1+\nu^{2}) + (1-\nu^{2})cos(\psi)} |
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39 | |
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40 | |
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41 | and the angle $\psi$ is defined as the orientation of the major axis of the |
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42 | ellipse with respect to the vector $\vec q$. The angle $\alpha$ is the angle |
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43 | between the axis of the cylinder and $\vec q$. |
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44 | |
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45 | |
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46 | Definition for 1D (no preferred orientation) |
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47 | -------------------------------------------- |
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48 | |
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49 | The form factor is averaged over all possible orientation before normalized |
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50 | by the particle volume |
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51 | |
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52 | .. math:: |
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53 | |
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54 | P(q) = \text{scale} <F^2> / V |
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55 | |
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56 | To provide easy access to the orientation of the elliptical cylinder, we |
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57 | define the axis of the cylinder using two angles $\theta$, $\phi$ and $\Psi$ |
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58 | (see :ref:`cylinder orientation <cylinder-angle-definition>`). The angle |
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59 | $\Psi$ is the rotational angle around its own long_c axis. |
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60 | |
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61 | All angle parameters are valid and given only for 2D calculation; ie, an |
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62 | oriented system. |
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63 | |
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64 | .. figure:: img/elliptical_cylinder_angle_definition.png |
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65 | |
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66 | Definition of angles for oriented elliptical cylinder, where axis_ratio is drawn >1, |
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67 | and angle $\Psi$ is now a rotation around the axis of the cylinder. |
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68 | |
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69 | .. figure:: img/elliptical_cylinder_angle_projection.png |
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70 | |
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71 | Examples of the angles for oriented elliptical cylinders against the |
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72 | detector plane, with $\Psi$ = 0. |
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73 | |
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74 | The $\theta$ and $\phi$ parameters to orient the cylinder only appear in the model when fitting 2d data. |
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75 | On introducing "Orientational Distribution" in the angles, "distribution of theta" and "distribution of phi" parameters will |
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76 | appear. These are actually rotations about the axes $\delta_1$ and $\delta_2$ of the cylinder, the $b$ and $a$ axes of the |
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77 | cylinder cross section. (When $\theta = \phi = 0$ these are parallel to the $Y$ and $X$ axes of the instrument.) |
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78 | The third orientation distribution, in $\psi$, is about the $c$ axis of the particle. Some experimentation may be required to |
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79 | understand the 2d patterns fully. (Earlier implementations had numerical integration issues in some circumstances when orientation |
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80 | distributions passed through 90 degrees, such situations, with very broad distributions, should still be approached with care.) |
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81 | |
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82 | NB: The 2nd virial coefficient of the cylinder is calculated based on the |
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83 | averaged radius $(=\sqrt{r_\text{minor}^2 * \text{axis ratio}})$ and length |
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84 | values, and used as the effective radius for $S(Q)$ when $P(Q)*S(Q)$ is applied. |
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85 | |
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86 | |
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87 | Validation |
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88 | ---------- |
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89 | |
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90 | Validation of our code was done by comparing the output of the 1D calculation |
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91 | to the angular average of the output of the 2D calculation over all possible |
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92 | angles. |
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93 | |
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94 | In the 2D average, more binning in the angle $\phi$ is necessary to get the |
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95 | proper result. The following figure shows the results of the averaging by |
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96 | varying the number of angular bins. |
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97 | |
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98 | .. figure:: img/elliptical_cylinder_averaging.png |
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99 | |
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100 | The intensities averaged from 2D over different numbers of bins and angles. |
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101 | |
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102 | References |
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103 | ---------- |
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104 | |
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105 | L A Feigin and D I Svergun, *Structure Analysis by Small-Angle X-Ray and |
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106 | Neutron Scattering*, Plenum, New York, (1987) [see table 3.4] |
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107 | |
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108 | Authorship and Verification |
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109 | ---------------------------- |
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110 | |
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111 | * **Author:** |
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112 | * **Last Modified by:** |
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113 | * **Last Reviewed by:** Richard Heenan - corrected equation in docs **Date:** December 21, 2016 |
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114 | |
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115 | """ |
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116 | |
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117 | from numpy import pi, inf, sqrt, sin, cos |
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118 | |
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119 | name = "elliptical_cylinder" |
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120 | title = "Form factor for an elliptical cylinder." |
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121 | description = """ |
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122 | Form factor for an elliptical cylinder. |
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123 | See L A Feigin and D I Svergun, Structure Analysis by Small-Angle X-Ray and Neutron Scattering, Plenum, New York, (1987). |
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124 | """ |
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125 | category = "shape:cylinder" |
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126 | |
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127 | # pylint: disable=bad-whitespace, line-too-long |
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128 | # ["name", "units", default, [lower, upper], "type","description"], |
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129 | parameters = [["radius_minor", "Ang", 20.0, [0, inf], "volume", "Ellipse minor radius"], |
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130 | ["axis_ratio", "", 1.5, [1, inf], "volume", "Ratio of major radius over minor radius"], |
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131 | ["length", "Ang", 400.0, [1, inf], "volume", "Length of the cylinder"], |
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132 | ["sld", "1e-6/Ang^2", 4.0, [-inf, inf], "sld", "Cylinder scattering length density"], |
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133 | ["sld_solvent", "1e-6/Ang^2", 1.0, [-inf, inf], "sld", "Solvent scattering length density"], |
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134 | ["theta", "degrees", 90.0, [-360, 360], "orientation", "cylinder axis to beam angle"], |
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135 | ["phi", "degrees", 0, [-360, 360], "orientation", "rotation about beam"], |
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136 | ["psi", "degrees", 0, [-360, 360], "orientation", "rotation about cylinder axis"]] |
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137 | |
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138 | # pylint: enable=bad-whitespace, line-too-long |
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139 | |
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140 | source = ["lib/polevl.c", "lib/sas_J1.c", "lib/gauss76.c", "lib/gauss20.c", |
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141 | "elliptical_cylinder.c"] |
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142 | |
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143 | demo = dict(scale=1, background=0, radius_minor=100, axis_ratio=1.5, length=400.0, |
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144 | sld=4.0, sld_solvent=1.0, theta=10.0, phi=20, psi=30, |
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145 | theta_pd=10, phi_pd=2, psi_pd=3) |
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146 | |
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147 | def ER(radius_minor, axis_ratio, length): |
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148 | """ |
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149 | Equivalent radius |
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150 | @param radius_minor: Ellipse minor radius |
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151 | @param axis_ratio: Ratio of major radius over minor radius |
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152 | @param length: Length of the cylinder |
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153 | """ |
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154 | radius = sqrt(radius_minor * radius_minor * axis_ratio) |
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155 | ddd = 0.75 * radius * (2 * radius * length |
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156 | + (length + radius) * (length + pi * radius)) |
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157 | return 0.5 * (ddd) ** (1. / 3.) |
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158 | |
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159 | def random(): |
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160 | import numpy as np |
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161 | # V = pi * radius_major * radius_minor * length; |
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162 | V = 10**np.random.uniform(3, 9) |
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163 | length = 10**np.random.uniform(1, 3) |
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164 | axis_ratio = 10**(4*np.random.uniform(-2, 2) |
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165 | radius_minor = np.sqrt(V/length/axis_ratio) |
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166 | Vf = 10**np.random.uniform(-4, -2) |
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167 | pars = dict( |
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168 | #background=0, sld=0, sld_solvent=1, |
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169 | scale=1e9*Vf/V, |
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170 | length=length, |
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171 | radius_minor=radius_minor, |
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172 | axis_ratio=axis_ratio, |
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173 | ) |
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174 | return pars |
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175 | |
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176 | q = 0.1 |
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177 | # april 6 2017, rkh added a 2d unit test, NOT READY YET pull #890 branch assume correct! |
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178 | qx = q*cos(pi/6.0) |
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179 | qy = q*sin(pi/6.0) |
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180 | |
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181 | tests = [ |
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182 | [{'radius_minor': 20.0, 'axis_ratio': 1.5, 'length':400.0}, 'ER', 79.89245454155024], |
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183 | [{'radius_minor': 20.0, 'axis_ratio': 1.2, 'length':300.0}, 'VR', 1], |
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184 | |
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185 | # The SasView test result was 0.00169, with a background of 0.001 |
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186 | [{'radius_minor': 20.0, 'axis_ratio': 1.5, 'sld': 4.0, 'length':400.0, |
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187 | 'sld_solvent':1.0, 'background':0.0}, |
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188 | 0.001, 675.504402], |
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189 | # [{'theta':80., 'phi':10.}, (qx, qy), 7.88866563001 ], |
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190 | ] |
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